Hooke's Law and Simple Harmonic Motion

We did an experiment with a vertical spring-mass system. Here is an example of data collected:
Mass------------T (period)
200 g-----------.3 s
400g------------.5 s

My question is why does the period increase as the mass increases?

I know that the amp. and T are independent of each other, but how do you show this mathematically? I now it’s easy to show visually, but are there any ties in deriving this, along with the relationship that T = 1/sqrt(x)?

Then, my teacher mentioned some of the following derivisions. I understand a majority of it, but I don’t know how these two parts directly correspond with each other. Does this look familiar to anyone? I don’t quite follow the M_efficiency line, but I do get the calculus.
1st Part
F = -kx, where T = 2*pi*sqrt(M/k) and Mg = kL (L=x)
M*[(d^2*x)/(dt^2)] = -(k/M)x (second derivative)
(d^2*x)/(dt^2) = -w^2*x, where w (or omega) = 2*pi*f
x(t) = Asin(wt + phi)
This is to show that T is proportional to k, I guess.